One-generated nilpotent Novikov algebras

One-generated nilpotent Novikov algebras 1 Luisa M. Camachoa, Iqboljon Karimjanovb,Ivan Kaygorodovc& Abror Khudoyberdiyevd aUniversity of Sevilla, Sevilla, Spain. bAndijan State University, Andijan, Uzbekistan. cCMCC, Universidade Federal do ABC, Santo Andr´e, Brazil. dNational University of Uzbekistan, Institute of Mathematics Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan. E-mail addresses: Luisa M. Camacho ([email protected]) Iqboljon Karimjanov (iqbo[email protected]) Ivan Kaygorodov (kaygorodov[email protected]) Abror Khudoyberdiyev ([email protected]) Abstract:We give a classification of 5- and 6-dimensional complex one-generated nilpotent Novikov algebras. Keywords:Novikov algebra, nilpotent algebra, algebraic classification, central extension. MSC2010: 17D25, 17A30. INTRODUCTION The algebraic classification (up to isomorphism) of algebras of dimension nfrom a certain variety defined by some family of polynomial identities is a classic problem in the theory of non-associative algebras. There are many results related to the algebraic classification of low-dimensional algebras in the varieties of Jordan, Lie, Leibniz, Zinbiel and many other algebras [1, 11, 14–18, 27–29, 32, 41, 42]. Another interesting direction is a study of one-generated objects. The description of onegenerated groups is well-known: there is only one one-generated group of order n. In the case of algebras, there are some similar results, such as the description of n-dimensional one-generated nilpotent associative [19], noncommutative Jordan [34], Leibniz and Zinbiel algebras [43]. It was proven that there is only one n-dimensional one-generated nilpotent algebra in these varieties. But on the other hand, as we can see in varieties of Novikov [36], assosymmetric [33], bicommutative [39], commutative [25], and terminal [38] algebras, there are more than one 4-dimensional one-generated nilpotent algebra from these varieties. It is easy to see that a nilpotent one-generated algebra is a central extension of a nilpotent one-generated algebra of a smaller dimension. In the present paper, we give an algebraic classification of 5- and 6-dimensional one-generated nilpotent Novikov algebras introduced by Novikov and Balinsky in 1985 [3]. The variety of Novikov algebras is defined by the following identities: (xy)z= (xz)y, (xy)z−x(yz) = (yx)z−y(xz). It contains the commutative associative algebras as a subvariety. On the other hand, the variety of Novikov algebras is the intersection of the variety of right commutative algebras (defined by the first Novikov identity) and the variety of left symmetric (Pre-Lie) algebras (defined by the second Novikov identity). Also, any Novikov algebra under the commutator multiplication is a Lie algebra [8,9], and Novikov algebras are related with Tortken and Novikov-Poisson algebras [6,20,47]. The systematic study of Novikov algebras started after the paper by Zelmanov where all complex finite-dimensional simple Novikov algebras were found [49]. The first nontrivial examples of infinite-dimensional simple Novikov algebras were constructed in [26]. Also, the simple Novikov algebras were described in infinite-dimensional case and over fields of positive characteristic [44, 46, 48]. The 1This work was supported by RFBR 18-31-20004; FAPESP 18/15712-0, 19/00192-3; Ministerio de Econom´ıa y Competitividad (Spain), grant MTM2016- 79661-P (European FEDER support included, UE). The authors thank Farukh Mashurov and the referee of the paper for constructive comments. 1 2 algebraic classification of 3-dimensional Novikov algebras was given in [4], and for some classes of 4-dimensional algebras, it was given in [7]; the geometric classification of 3-dimensional Novikov algebras was given in [5] and of 4-dimensional nilpotent Novikov algebras in [35]. Many other pure algebraic properties were studied in a series of papers by Dzhumadildaev [21–24]. Our method to classify nilpotent Novikov algebras is based on the calculation of central extensions of smaller nilpotent algebras from the same variety. The algebraic study of central extensions of Lie and non-Lie algebras has been an important topic for years [2, 30, 31, 36, 40, 45, 50]. First, Skjelbred and Sund used central extensions of Lie algebras for a classification of nilpotent Lie algebras [45]. After that, using the method described by Skjelbred and Sund, all non-Lie central extensions of all 4-dimensional Malcev algebras were described [31], and also all non-associative central extensions of 3-dimensional Jordan algebras [30], all anticommutative central extensions of the 3-dimensional anticommutative algebras [10], and all central extensions of the 2-dimensional algebras [12]. Note that the method of central extensions is an important tool in the classification of nilpotent algebras, which was used to describe all 4-dimensional nilpotent associative algebras [18], all 5-dimensional nilpotent Jordan algebras [29], all 5-dimensional nilpotent restricted Lie algebras [16], all 5-dimensional nilpotent associative commutative algebras [39], all 6-dimensional nilpotent Lie algebras [15, 17], all 6-dimensional nilpotent Malcev algebras [32], all 6-dimensional nilpotent anticommutative algebras [37], all 8-dimensional dual Mock Lie algebras [13], and some others. 1. THE ALGEBRAIC CLASSIFICATION OF NILPOTENT NOVIKOV ALGEBRAS 1.1. Method of classification of nilpotent algebras. Throughout this paper, we use the notations and methods well written in [12, 30, 31], which we have adapted for the Novikov case with some modifications. Further in this section we give some important definitions. Let (A,·)be a Novikov algebra over Cand Va vector space over C. The C-linear space Z2(A,V)is defined as the set of all bilinear maps θ:A×A−→ Vsuch that θ(xy, z) = θ(xz, y), θ(xy, z)−θ(x, yz) = θ(yx, z)−θ(y, xz). These elements will be called cocycles. For a linear map ffrom Ato V, if we define δf :A×A−→ Vby δf(x, y) = f(xy), then δf ∈Z2(A,V). We define B2(A,V) = {θ=δf :f∈Hom (A,V)}. We define the second cohomology space H2(A,V) as the quotient space Z2(A,V)B2(A,V). Let Aut(A)be the automorphism group of Aand let φ∈Aut(A). For θ∈Z2(A,V)define the action of the group Aut(A) on H2(A,V)by φθ(x, y) = θ(φ(x), φ (y)). It is easy to verify that B2(A,V)is invariant under the action of Aut(A).So, we have an induced action of Aut(A)on H2(A,V). Let Abe a Novikov algebra of dimension mover Cand Vbe a C-vector space of dimension k. For θ∈Z2(A,V), define on the linear space Aθ=A⊕Vthe bilinear product “ [−,−]Aθ” by [x+x′, y +y′]Aθ=xy +θ(x, y)for all x, y ∈A, x′, y′∈V. The algebra Aθis called an k-dimensional central extension of Aby V. One can easily check that Aθis a Novikov algebra if and only if θ∈Z2(A,V). Call the set Ann(θ) = {x∈A:θ(x, A) + θ(A, x) = 0}the annihilator of θ. We recall that the annihilator of an algebra Ais defined as the ideal Ann(A) = {x∈A:xA+Ax= 0}. Observe that Ann (Aθ) = (Ann(θ)∩Ann(A)) ⊕V. The following result shows that every algebra with a non-zero annihilator is a central extension of a smaller-dimensional algebra. As a consequence, it is easy to see that a one-generated nilpotent algebra is a central extension of a one-generated nilpotent algebra of a smaller dimension. Lemma 1. Let Abe an n-dimensional Novikov algebra such that dim(Ann(A)) = m6= 0. Then there exists, up to isomorphism, a unique (n−m)-dimensional Novikov algebra A′and a bilinear map θ∈Z2(A,V)with Ann(A)∩Ann(θ) = 0, where Vis a vector space of dimension m, such that A∼ =A′θand A/Ann(A)∼ =A′. Proof. Let A′be a linear complement of Ann(A)in A. Define a linear map P:A−→ A′by P(x+v) = xfor x∈A′and v∈Ann(A), and define a multiplication on A′by [x, y]A′=P(xy)for x, y ∈A′. For x, y ∈A, we have P(xy) = P((x−P(x) + P(x))(y−P(y) + P(y))) = P(P(x)P(y)) = [P(x), P(y)]A′. 3 Since Pis a homomorphism P(A) = A′is a Novikov algebra and A/Ann(A)∼ =A′, which gives us the uniqueness. Now, define the map θ:A′×A′−→ Ann(A)by θ(x, y) = xy −[x, y]A′. Thus, A′ θis Aand therefore θ∈Z2(A,V)and Ann(A)∩Ann(θ) = 0. Definition 2. Let Abe an algebra and Ibe a subspace of Ann(A). If A=A0⊕Ithen Iis called an annihilator component of A. Definition 3. A central extension of an algebra Awithout annihilator component is called a non-split central extension. Our task is to find all central extensions of an algebra Aby a space V. In order to solve the isomorphism problem we need to study the action of Aut(A)on H2(A,V). To do that, let us fix a basis e1,...,esof V, and θ∈Z2(A,V). Then θcan be uniquely written as θ(x, y) = s X i=1 θi(x, y)ei, where θi∈Z2(A,C). Moreover, Ann(θ) = Ann(θ1)∩Ann(θ2)∩· · ·∩Ann(θs). Furthermore, θ∈B2(A,V)if and only if all θi∈B2(A,C). It is not difficult to prove (see [31, Lemma 13]) that given a Novikov algebra Aθ, if we write as above θ(x, y) = s X i=1 θi(x, y)ei∈Z2(A,V)and Ann(θ)∩Ann (A) = 0, then Aθhas an annihilator component if and only if [θ1],[θ2],...,[θs]are linearly dependent in H2(A,C). Let Vbe a finite-dimensional vector space over C. The Grassmannian Gk(V)is the set of all k-dimensional linear subspaces of V. Let GsH2(A,C)be the Grassmannian of subspaces of dimension sin H2(A,C). There is a natural action of Aut(A)on GsH2(A,C). Let φ∈Aut(A). For W=h[θ1],[θ2],...,[θs]i ∈ GsH2(A,C)define φW =h[φθ1],[φθ2],...,[φθs]i. We denote the orbit of W∈GsH2(A,C)under the action of Aut(A)by Orb(W). Given W1=h[θ1],[θ2],...,[θs]i, W2=h[ϑ1],[ϑ2],...,[ϑs]i ∈ GsH2(A,C), we easily have that if W1=W2, then s T i=1 Ann(θi)∩Ann (A) = s T i=1 Ann(ϑi)∩Ann(A), and therefore we can introduce the set Ts(A) = (W=h[θ1],[θ2],...,[θs]i ∈ GsH2(A,C): s \ i=1 Ann(θi)∩Ann(A) = 0), which is stable under the action of Aut(A). Now, let Vbe an s-dimensional linear space and let us denote by E(A,V)the set of all non-split s-dimensional central extensions of Aby V. By above, we can write E(A,V) = (Aθ:θ(x, y) = s X i=1 θi(x, y)eiand h[θ1],[θ2],...,[θs]i ∈ Ts(A)). We also have the following result, which can be proved as in [31, Lemma 17]. Lemma 4. Let Aθ,Aϑ∈E(A,V). Suppose that θ(x, y) = s X i=1 θi(x, y)eiand ϑ(x, y) = s X i=1 ϑi(x, y)ei. Then the Novikov algebras Aθand Aϑare isomorphic if and only if Orb h[θ1],[θ2],...,[θs]i= Orb h[ϑ1],[ϑ2],...,[ϑs]i. This shows that there exists a one-to-one correspondence between the set of Aut(A)-orbits on Ts(A)and the set of isomorphism classes of E(A,V). Consequently we have a procedure that allows us, given a Novikov algebra A′of dimension n−s, to construct all non-split central extensions of A′. This procedure is: Procedure 4 (1) For a given Novikov algebra A′of dimension n−s, determine H2(A′,C),Ann(A′)and Aut(A′). (2) Determine the set of Aut(A′)-orbits on Ts(A′). (3) For each orbit, construct the Novikov algebra associated with a representative of it. 1.2. Notations. Let Abe a Novikov algebra with a basis e1, e2,...,en. Then by ∆ij we will denote the bilinear form ∆ij :A× A−→ Cwith ∆ij (el, em) = δilδjm. Then the set {∆ij : 1 ≤i, j ≤n}is a basis for the linear space of the bilinear forms on A. Then every bilinear form θcan be uniquely written as θ=X 1≤i,j≤n cij∆ij, where cij ∈C. Let us fix the following notations: Ni j—jth i-dimensional one-generated Novikov algebra. 1.3. Low dimensional one-generated nilpotent Novikov algebras. Thanks to [35] we have the algebraic classification of 2-, 3- and 4- dimensional and 5-dimensional with 2-dimensional annihilator one-generated nilpotent algebras. Let us give a list if 1-, 2-, 3- and 4-dimensional one-generated nilpotent Novikov algebras. N2 01 :e1e1=e2 N3 01 :e1e1=e2e2e1=e3 N3 02(λ) : e1e1=e2e1e2=e3e2e1=λe3 N4 01 :e1e1=e2e1e2=e3e2e1=e4 N4 02 :e1e1=e2e2e1=e3e1e3=e4e3e1=−e4 N4 03(λ) : e1e1=e2e1e2=e3e1e3= (2 −λ)e4e2e1=λe3e2e2=λe4e3e1=λe4 N4 04 :e1e1=e2e1e2=e3e1e3= 2e4e2e1=e4 N4 05 :e1e1=e2e1e2=e3e1e3=e4e2e1=e3+e4e2e2=e4e3e1=e4 Remark 5. Note that a non-split central extension of a split algebra can not be a one-generated algebra. Hence, we will consider central extensions only of non-split one-generated nilpotent algebras. 2. CLASSIFICATION OF 5-DIMENSIONAL ONE-GENERATED NILPOTENT NOVIKOV ALGEBRAS All the necessary information about coboundaries, cocycles and second cohomology spaces of 5-dimensional one-generated nilpotent Novikov algebras is given in Table A (see Table A from Appendix). From the results of [35], it is not difficult to see that there are two 5-dimensional one-generated Novikov algebras with 2-dimensional annihilator: N5 01 :e1e1=e2e1e2=e4e1e3=e5e2e1=e3e3e1=−e5 N5 02(λ):e1e1=e2e1e2=e3e1e3= (2 −λ)e5e2e1=λe3+e4e2e2=λe5e3e1=λe5 2.1. Central extensions of N4 01.Let us use the following notations ∇1= [∆13],∇2= [∆14]−[∆41],∇3= [∆22] + [∆31] + 2[∆41]. The automorphism group of N4 01 consists of invertible matrices of the form φ=    x0 0 0 y x20 0 z xy x30 v xy 0x3     . Since φT    0 0 α1α2 0α30 0 α30 0 0 2α3−α20 0 0     φ=    α∗α∗∗ α∗ 1α∗ 2 α∗∗∗ α∗ 30 0 α∗ 30 0 0 2α∗ 3−α∗ 20 0 0     , 5 we have that the action of Aut(N4 01)on the subspace h 3 P i=1 αi∇iiis given by h 3 P i=1 α∗ i∇ii,where α∗ 1=x4α1, α∗ 2=x4α2, α∗ 3=x4α3. 2.1.1. 1-dimensional central extensions of N4 01.Since we are interested only in new algebras, we have the following cases: (1) if α36= 0,then we have the representative hλ∇1+µ∇2+∇3i; (2) if α3= 0,then (α1, α2)6= (0,0) and we have the representative h∇1+λ∇2iwith λ6= 0. Hence, we have the following new 5-dimensional algebras: N5 03(λ, µ):e1e1=e2e1e2=e3e1e3=λe5e1e4=µe5 e2e1=e4e2e2=e5e3e1=e5e4e1= (2 −µ)e5 N5 04(λ)λ6=0 :e1e1=e2e1e2=e3e1e3=e5e1e4=λe5e2e1=e4e4e1=−λe5 2.1.2. 2-dimensional central extensions of N4 01.Consider the vector space generated by the following two cocycles θ1=α1∇1+α2∇2+α3∇3, θ2=β1∇1+β2∇2. If α3= 0,then we have the representative h∇1,∇2i.If α36= 0,then (1) β2= 0,then we have the family of representatives h∇1, λ∇2+∇3i; (2) β26= 0,then we have the family of representatives hλ∇1+∇2, µ∇1+∇3i. Hence, we have the following 6-dimensional algebras: N6 01 :e1e1=e2e1e2=e3e1e3=e5e1e4=e6e2e1=e4e4e1=−e6 N6 02(λ):e1e1=e2e1e2=e3e1e3=e5e1e4=λe6 e2e1=e4e2e2=e6e3e1=e6e4e1= (2 −λ)e6 N6 03(λ, µ):e1e1=e2e1e2=e3e1e3=λe5+µe6e1e4=e5 e2e1=e4e2e2=e6e3e1=e6e4e1=−e5+ 2e6 2.2. Central extensions of N4 02.Let us use the following notations ∇1= [∆12],∇2= 2[∆14]−[∆23]−[∆41]. The automorphism group of N4 02 consists of invertible matrices of the form φ=    x000 0x20 0 y0x30 z0 0 x4     . Since φT    0α10 2α2 0 0 −α20 0 0 0 0 −α20 0 0     φ=    α∗α∗ 10 2α∗ 2 α∗∗ 0−α∗ 20 0 0 0 0 −α∗ 20 0 0     , we have that the action of Aut(N4 02)on the subspace h 2 P i=1 αi∇iiis given by h 2 P i=1 α∗ i∇ii,where α∗ 1=x3α1, α∗ 2=x5α2. 2.2.1. 1-dimensional central extensions of N4 02.Since we are interested only in new algebras, we have assume α26= 0,and we have the following cases: (1) if α1= 0,then we have the representative h∇2i; (2) if α16= 0,then by choosing x=pα1/α2,we have the representative h∇1+∇2i. 6 Thus, we have the following 5-dimensional algebras: N5 05 :e1e1=e2e1e3=e4e1e4= 2e5e2e1=e3e2e3=−e5e3e1=−e4e4e1=−e5 N5 06 :e1e1=e2e1e2=e5e1e3=e4e1e4= 2e5e2e1=e3e2e3=−e5e3e1=−e4e4e1=−e5 2.2.2. 2-dimensional central extensions of N4 02.Since dim H2(N4 02)= 2,there is only one 2-dimensional central extension of N4 02.Thus, we have the following 6-dimensional algebra: N6 04 :e1e1=e2e1e2=e5e1e3=e4e1e4= 2e6e2e1=e3e2e3=−e6e3e1=−e4e4e1=−e6 2.3. Central extensions of N4 03(λ)λ/∈{0,1}.Let us use the following notations ∇1= [∆21],∇2= (3 −2λ)[∆14] + λ(2 −λ)[∆23] + λ[∆32] + λ[∆41]. The automorphism group of N4 03(λ)consists of invertible matrices of the form φ=    x0 0 0 0x20 0 y0x30 z2xy 0x4     . Since φT    0 0 0 (3 −2λ)α2 α10λ(2 −λ)α20 0λα20 0 λα20 0 0     φ=    α∗∗ α∗0 (3 −2λ)α∗ 2 α∗ 1+λα∗0λ(2 −λ)α∗ 20 0λα∗ 20 0 λα∗ 20 0 0     , we have that the action of Aut(N4 03(λ)) on the subspace h 2 P i=1 αi∇iiis given by h 2 P i=1 α∗ i∇ii,where α∗ 1=x2(xα1+ 2λ(λ−1)yα2), α∗ 2=x5α2. 2.3.1. 1-dimensional central extensions of N4 03(λ)λ/∈{0,1}.Since we are interested only in new algebras, we have α26= 0. Then choosing y=xα1 2λ(1−λ)α2,we have the representative h∇2i.Hence, we have the following 5-dimensional algebras: N5 07(λ)λ/∈{0,1}:e1e1=e2e1e2=e3e1e3= (2 −λ)e4e1e4= (3 −2λ)e5e2e1=λe3 e2e2=λe4e2e3=λ(2 −λ)e5e3e1=λe4e3e2=λe5e4e1=λe5 2.3.2. 2-dimensional central extensions of N4 03(λ)λ/∈{0,1}.Since dim H2(N4 03(λ)λ6=0;1)= 2,there is only one 2-dimensional central extension of N4 03(λ)λ/∈{0,1}.Hence, we have the following 6-dimensional algebras: N6 05(λ)λ/∈{0,1}:e1e1=e2e1e2=e3e1e3= (2 −λ)e4e1e4= (3 −2λ)e5e2e1=λe3+e6 e2e2=λe4e2e3=λ(2 −λ)e5e3e1=λe4e3e2=λe5e4e1=λe5 2.4. Central extensions of N4 03(0).Let us use the following notations ∇1= [∆14],∇2= [∆21],∇3= 2[∆23]−2[∆32] + [∆41]. The automorphism group of N4 03(0) consists of invertible matrices of the form φ=    x0 0 0 y x20 0 z xy x30 t2xz 2x2y x4     . 7 Since φT    0 0 0 α1 α20 2α30 0−2α30 0 α30 0 0     φ=    α∗α∗∗ α∗∗∗ α∗ 1 α∗ 20 2α∗ 30 0−2α∗ 30 0 α∗ 30 0 0     , we have that the action of Aut(N4 03(0)) on the subspace h 3 P i=1 αi∇iiis given by h 3 P i=1 α∗ i∇ii,where α∗ 1=x5α1, α∗ 2=x3α2+ (4x2z−2xy2)α3, α∗ 3=x5α3. 2.4.1. 1-dimensional central extensions of N4 03(0).Since we are interested only in new algebras, we have (α1, α3)6= (0,0) and: (1) if α3= 0, α2= 0,then we have the representative h∇1i; (2) if α3= 0, α26= 0,then choosing x=pα2/α1,we have the representative h∇1+∇2i; (3) if α36= 0,then choosing y= 0, z =−α2 4α35 √α3,we have the family of representatives hλ∇1+∇3i. Hence, we have the following 5-dimensional algebras: N5 07(0) :e1e1=e2e1e2=e3e1e3= 2e4e1e4= 3e5 N5 08 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e1=e5 N5 09(λ):e1e1=e2e1e2=e3e1e3= 2e4e1e4=λe5e2e3= 2e5e3e2=−2e5e4e1=e5 2.4.2. 2-dimensional central extensions of N4 03(0).Consider the vector space generated by the following two cocycles θ1=α1∇1+α2∇2+α3∇3, θ2=β1∇1+β2∇2. If α3= 0,then we have the representative h∇1,∇2i.If α36= 0,then (1) if β1= 0, β26= 0,then we have the family of representatives hλ∇1+∇3,∇2i; (2) if β16= 0, β26= 0,then choosing x=pβ2/β1, y = 0, z =x3α1−xα2 4α3,we have the representative h∇1+∇2,∇3i; (3) if β16= 0, β2= 0,then we have the representative h∇1,∇3i. Hence we have the following 6-dimensional algebras: N6 05(0) :e1e1=e2e1e2=e3e1e3= 2e4e1e4= 3e5e2e1=e6 N6 06(λ):e1e1=e2e1e2=e3e1e3= 2e4e1e4=λe5e2e1=e6e2e3= 2e5e3e2=−2e5e4e1=e5 N6 07 :e1e1=e2e1e2=e3e1e3= 2e4e1e4=e5e2e1=e5e2e3= 2e6e3e2=−2e6e4e1=e6 N6 08 :e1e1=e2e1e2=e3e1e3= 2e4e1e4=e5e2e3= 2e6e3e2=−2e6e4e1=e6 2.5. Central extensions of N4 03(1).Let us use the following notations ∇1= [∆21],∇2= [∆14] + [∆23] + [∆32] + [∆41]. The automorphism group of N4 03(1) consists of invertible matrices of the form φ=    x0 0 0 y x20 0 z2xy x30 t2xz +y23x2y x4     . Since φT    000α2 α10α20 0α20 0 α20 0 0     φ=    α∗∗∗ α∗∗ α∗α∗ 2 α∗ 1+α∗∗ α∗α∗ 20 α∗α∗ 20 0 α∗ 20 0 0     , 8 we have that the action of Aut(N4 03(1)) on the subspace h 2 P i=1 αi∇iiis given by h 2 P i=1 α∗ i∇ii,where α∗ 1=x3α1, α∗ 2=x5α2. 2.5.1. 1-dimensional central extensions of N4 03(1).We may assume α26= 0,so we have the following cases: (1) if α1= 0,then we have the representative h∇2i; (2) if α16= 0,then choosing x=pα1/α2we have the representative h∇1+∇2i. Hence, we have the following 5-dimensional algebras: N5 07(1) :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e1=e3 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 N5 10 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e1=e3+e5 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 2.5.2. 2-dimensional central extensions of N4 03(1).Since dim H2(N4 03(1))= 2,there is only one 2-dimensional central extension of N4 03(1).Hence, we have the following 6-dimensional algebra: N6 05(1) :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e1=e3+e6 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 2.6. Central extensions of N4 04.Let us use the following notations ∇1= [∆13],∇2= 2[∆14] + [∆22] + [∆31],∇3= [∆14]−2[∆23] + 2[∆32]−[∆41]. The automorphism group of N4 04 consists of invertible matrices of the form φ=    1 0 0 0 x1 0 0 y x 1 0 z x + 2y2x1     . Since φT    0 0 α12α2+α3 0α2−2α30 α22α30 0 −α30 0 0     φ=    α∗∗∗ α∗∗ 2α∗+α∗ 12α∗ 2+α∗ 3 α∗α∗ 2−2α∗ 30 α∗ 22α∗ 30 0 −α∗ 30 0 0     , we have that the action of Aut(N4 04)on the subspace h 3 P i=1 αi∇iiis given by h 3 P i=1 α∗ i∇ii,where α∗ 1=α1+ 2(x−2x2+ 4y)α3, α∗ 2=α2, α∗ 3=α3. 2.6.1. 1-dimensional central extensions of N4 04.Since we are interested only in new algebras, we assume (α2, α3)6= (0,0) and we have the following cases: (1) if α26= 0, α3= 0,then we have the family of representatives hλ∇1+∇2i; (2) if α36= 0,then by choosing y=−α1+(2x−4x2)α3 8α3,we have the family of representatives hλ∇2+∇3i. Hence, we have the following 5-dimensional algebras: N5 11(λ):e1e1=e2e1e2=e3e1e3= 2e4+λe5e1e4= 2e5e2e1=e4e2e2=e5e3e1=e5 N5 12(λ):e1e1=e2e1e2=e3e1e3= 2e4e1e4= (2λ+ 1)e5e2e1=e4 e2e2=λe5e2e3=−2e5e3e1=λe5e3e2= 2e5e4e1=−e5 9 2.6.2. 2-dimensional central extensions of N4 04.Consider the vector space generated by the following two cocycles θ1=α1∇1+α2∇2+α3∇3, θ2=β1∇1+β2∇2. If α3= 0,then we have the representative h∇1,∇2i.If α36= 0,then (1) if β2= 0,then we have the family of representatives h∇1, λ∇2+∇3i; (2) if β26= 0,then we have the family of representatives hλ∇1+∇2,∇3i. Hence, we have the following 6-dimensional algebras: N6 09 :e1e1=e2e1e2=e3e1e3= 2e4+e6e1e4= 2e5e2e1=e4e2e2=e5e3e1=e5 N6 10(λ):e1e1=e2e1e2=e3e1e3= 2e4+e6e1e4= (2λ+ 1)e5e2e1=e4 e2e2=λe5e2e3=−2e5e3e1=λe5e3e2= 2e5e4e1=−e5 N6 11(λ):e1e1=e2e1e2=e3e1e3= 2e4+λe6e1e4=e5+ 2e6e2e1=e4 e2e2=e6e2e3=−2e5e3e1=e6e3e2= 2e5e4e1=−e5 2.7. Central extensions of N4 05.Let us use the following notations ∇1= [∆21],∇2=−2[∆13] + [∆14] + [∆23] + [∆32] + [∆41]. The automorphism group of N4 05 consists of invertible matrices of the form φ=    1 0 0 0 x1 0 0 y2x1 0 z x2+x+ 2y3x1     . Since φT    0 0 −2α2α2 α10α20 0α20 0 α20 0 0     φ=    α∗∗∗ α∗∗ −2α∗ 2+α∗α∗ 2 α∗ 1+α∗+α∗∗ α∗α∗ 20 α∗α∗ 20 0 α∗ 20 0 0     , we have that the action of Aut(N4 05)on the subspace h 2 P i=1 αi∇iiis given by h 2 P i=1 α∗ i∇ii,where α∗ 1=α1, α∗ 2=α2. 2.7.1. 1-dimensional central extensions of N4 05.Assuming α26= 0,we have the family of representatives hλ∇1+∇2i. Hence, we have the following 5-dimensional algebras: N5 13(λ):e1e1=e2e1e2=e3e1e3=e4−2e5e1e4=e5e2e1=e3+e4+λe5 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 2.7.2. 2-dimensional central extensions of N4 05.Since dim H2(N4 05)= 2,there is only one 2-dimensional central extension of N4 05.Hence, we have the following 6-dimensional algebra: N6 12 :e1e1=e2e1e2=e3e1e3=e4−2e5e1e4=e5e2e1=e3+e4+e6 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 Summarizing the results of the present sections and using the information from Table A from Appendix, we have the following theorem. Theorem 6. Let Nbe a 5-dimensional complex one-generated nilpotent Novikov algebra. Then Nis isomorphic to one of the algebras N5 j, j = 01,...,13 listed in Table B presented in Appendix. 16 3.9. Central extensions of N5 07(0).Let us use the following notations ∇1= [∆15],∇2= [∆21],∇3= 2[∆23]−2[∆32] + [∆41]. The automorphism group of N5 07(0) consists of invertible matrices of the form φ=      x0 0 0 0 y x20 0 0 z xy x30 0 v2xz 2x2y x40 w3xv 6x2z3x3y x5       . Since φT      0 0 0 0 α1 α20 2α30 0 0−2α30 0 0 α30 0 0 0 0 0 0 0 0       φ=      α∗∗∗∗ α∗∗∗ α∗∗ α∗α∗ 1 α∗ 20 2α∗ 30 0 0−2α∗ 30 0 0 α∗ 30 0 0 0 0 0 0 0 0       , we have that the action of Aut(N5 07(0)) on the subspace h 3 P i=1 αi∇iiis given by h 3 P i=1 α∗ i∇ii,where α∗ 1=x6α1, α∗ 2=x3α2+ 2x(2xz −y2)α3, α∗ 3=x5α3. Assuming α16= 0, we have the following cases: (1) if α36= 0,then choosing x=α3 α1, y = 0, z =−xα2 4α3,we have the representative h∇1+∇3i; (2) if α3= 0, α26= 0,then choosing x=3 pα2/α1,we have the representative h∇1+∇2i; (3) if α3=α2= 0,then we have the representative h∇1i. Hence, we have the following new algebras: N6 31 :e1e1=e2e1e2=e3e1e3= 2e4e1e4= 3e5e1e5=e6 e2e3= 2e6e3e2=−2e6e4e1=e6 N6 32 :e1e1=e2e1e2=e3e1e3= 2e4e1e4= 3e5e1e5=e6e2e1=e6 N6 29(0) :e1e1=e2e1e2=e3e1e3= 2e4e1e4= 3e5e1e5= 4e6 3.10. Central extensions of N5 07(1).Let us use the following notations ∇1= [∆21],∇2= [∆15] + [∆24] + [∆33] + [∆42] + [∆51]. The automorphism group of N5 07(1) consists of invertible matrices of the form φ=      x0 0 0 0 y x20 0 0 z2xy x30 0 v2xz +y23x2y x40 w2xv + 2yz 3x2z+ 3xy24x3y x5       . Since φT      0 0 0 0 α2 α10 0 α20 0 0 α20 0 0α20 0 0 α20 0 0 0       φ=      α∗∗∗∗ α∗∗∗ α∗∗ α∗α∗ 2 α∗ 1+α∗∗∗ α∗∗ α∗α∗ 20 α∗∗ α∗α∗ 20 0 α∗α∗ 20 0 0 α∗ 20 0 0 0       , 17 we have that the action of Aut(N5 07(1)) on the subspace h 2 P i=1 αi∇iiis given by h 2 P i=1 α∗ i∇ii,where α∗ 1=x3α1, α∗ 2=x6α2. Assuming α26= 0, we have the following cases: (1) if α1= 0,then we have the representative h∇2i; (2) if α16= 0,then choosing x=3 pα1/α2, we have the representative h∇1+∇2i. Hence, we have the algebras: N6 29(1) :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e1e5=e6 e2e1=e3e2e2=e4e2e3=e5e2e4=e6e3e1=e4 e3e2=e5e3e3=e6e4e1=e5e4e2=e6e5e1=e6 N6 33 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e1e5=e6 e2e1=e3+e6e2e2=e4e2e3=e5e2e4=e6e3e1=e4 e3e2=e5e3e3=e6e4e1=e5e4e2=e6e5e1=e6 3.11. Central extensions of N5 08.Let us use the following notations ∇1= [∆21],∇2= 2[∆15] + [∆22] + [∆31],∇3= [∆23]−[∆32] + [∆41]. The automorphism group of N5 08 consists of invertible matrices of the form φ1=      1 0 0 0 0 x1 0 0 0 y x 1 0 0 z y x 1 0 v x +z y x 1       , φ2=      −1 0 0 0 0 x1 0 0 0 y−x−1 0 0 z−y x 1 0 v−x−z y −x−1       . Since φT i      0 0 0 0 2α2 α1α2α30 0 α2−α30 0 0 α30 0 0 0 0 0 0 0 0       φi=      α∗∗∗∗ α∗∗∗ α∗∗ α∗2α∗ 2 α∗ 1+α∗α∗ 2α∗ 30 0 α∗ 2−α∗ 30 0 0 α∗ 30 0 0 0 0 0 0 0 0       , we have that the action of Aut(N5 08)on the subspace h 3 P i=1 αi∇iiis given by h 3 P i=1 α∗ i∇ii,where for i= 1 : α∗ 1=α1−(x2−2y)α3, α∗ 2=α2, α∗ 3=α3, for i= 2 : α∗ 1=−α1+ (x2+ 2y)α3, α∗ 2=α2, α∗ 3=−α3. Assuming α26= 0 and using φ1we have the following cases: (1) if α3= 0,then we have the family of representatives hλ∇1+∇2i; (2) if α36= 0,then choosing y=x2α3−α1 2α3,we have the family of representatives h∇2+λ∇3iλ6=0. Hence, we have the following new algebras: N6 34(λ):e1e1=e2e1e2=e3e1e3=e4e1e4=e5 e1e5= 2e6e2e1=e5+λe6e2e2=e6e3e1=e6 N6 35(λ)λ6=0 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5 e1e5= 2e6e2e1=e5e2e2=e6e2e3=λe6 e3e1=e6e3e2=−λe6e4e1=λe6 Moreover, using the action of φ2we have that N6 34(λ)∼ =N6 34(−λ)and N6 35(λ)∼ =N6 35(−λ). 18 3.12. Central extensions of N5 10.Let us use the following notations ∇1= [∆12],∇2=−2[∆13] + [∆15] + [∆24] + [∆33] + [∆42] + [∆51]. The automorphism group of N5 10 consists of invertible matrices of the form φ1=      1 0 0 0 0 x1 0 0 0 y2x1 0 0 z x2+ 2y3x1 0 v2xy +x+ 2z3x2+ 3y4x1       , φ2=      −1 0 0 0 0 x1 0 0 0 y−2x−1 0 0 z x2−2y3x1 0 v2xy −x−2z−3x2+ 3y−4x−1       . Since φT i      0α1−2α20α2 0 0 0 α20 0 0 α20 0 0α20 0 0 α20 0 0 0       φi=      α∗∗∗∗ α∗ 1+α∗∗∗ −2α∗ 2+α∗∗ α∗α∗ 2 α∗∗∗ α∗∗ α∗α∗ 20 α∗∗ α∗α∗ 20 0 α∗α∗ 20 0 0 α∗ 20 0 0 0       , we have that the action of Aut(N5 10)on the subspace h 2 P i=1 αi∇iiis given by h 2 P i=1 α∗ i∇ii,where α∗ 1= (−1)iα1−4xα2, α∗ 2=α2. It is easy to see that we have only one non-trivial orbit with representative h∇2i.The corresponding algebra is N6 36 :e1e1=e2e1e2=e3e1e3=e4−2e6e1e4=e5e1e5=e6 e2e1=e3+e5e2e2=e4e2e3=e5e2e4=e6e3e1=e4 e3e2=e5e3e3=e6e4e1=e5e4e2=e6e5e1=e6 3.13. Central extensions of N5 11(λ).Let us use the following notations ∇1= [∆13],∇2= [∆15] + [∆32],∇3= [∆14]−2[∆23] + 2[∆32]−[∆41]. The automorphism group of N5 11(λ)consists of invertible matrices of the form φ=      1 0 0 0 0 x1 0 0 0 y x 1 0 0 z x + 2y2x1 0 v x2+ (λ+ 1)y+ 2z(λ+ 3)x+ 4y2x1       . Since φT      0 0 α1α3α2 0 0 −2α30 0 0α2+ 2α30 0 0 −α30 0 0 0 0 0 0 0 0       φ=      α∗∗∗∗ α∗∗∗ α∗ 1+ 2α∗∗ +λα∗α∗ 3+ 2α∗α∗ 2 α∗∗ α∗−2α∗ 30 0 α∗α∗ 2+ 2α∗ 30 0 0 −α∗ 30 0 0 0 0 0 0 0 0       , we have that the action of Aut(N5 11(λ)) on the subspace h 3 P i=1 αi∇iiis given by h 3 P i=1 α∗ i∇ii,where α∗ 1=α1+ (3x+ 4y−2x2)α2+ (8y+ 2x−4x2)α3, α∗ 2=α2, α∗ 3=α3. Assuming α26= 0,we have the follofing cases (1) if α26=−2α3,then choosing y=−α1+x(3α2+2α3)−2x2(α2+2α3) 4(α2+2α3),we have the family of orbits with the representatives h∇2+µ∇3iµ6=−1 2; 19 (2) if α2=−2α3,then choosing x=−α1 2α2,we have the family of orbits with the representatives h∇2−1 2∇3i. Thus, we obtain the family of algebras: N6 37(λ, µ):e1e1=e2e1e2=e3e1e3= 2e4+λe5e1e4= 2e5+µe6 e1e5=e6e2e1=e4e2e2=e5e2e3=−2µe6 e3e1=e5e3e2= (1 + 2µ)e6e4e1=−µe6 3.14. Central extensions of N5 13(λ).Let us use the following notations ∇1= [∆21],∇2= (1 −2λ)[∆13]−2[∆14] + [∆15]−[∆23] + [∆24] + [∆33] + [∆41] + [∆42] + [∆51]. The automorphism group of N5 13(λ)consists of invertible matrices of the form φ=      1 0 0 0 0 x1 0 0 0 y2x1 0 0 z x2+x+ 2y3x1 0 v λx + 2xy −2y+ 2z3x2−3x+ 3y4x1       . Since φT      0α1(1 −2λ)α2−2α2α2 0 0 −α2α20 0 0 α20 0 α2α20 0 0 α20 0 0 0       φ=       α∗∗∗∗ α∗ 1+α∗∗∗ (1 −2λ)α∗ 2+α∗∗ −2α∗−2α∗ 2+α∗α∗ 2 α∗∗∗ +α∗∗ +λα∗α∗∗ −α∗ 2+α∗α∗ 20 α∗∗ α∗α∗ 20 0 α∗ 2+α∗α∗ 20 0 0 α∗ 20 0 0 0       , we have that the action of Aut(N5 13(λ)) on the subspace h 2 P i=1 αi∇iiis given by h 2 P i=1 α∗ i∇ii,where α∗ 1=α1−(λx −x2+x+y)α2, α∗ 2=α2. It is easy to see that we have only one non-trivial orbit with the representative h∇2iand obtain the family of algebras: N6 38(λ):e1e1=e2e1e2=e3e1e3=e4−2e5+ (1 −2λ)e6e1e4=e5−2e6e1e5=e6 e2e1=e3+e4+λe5e2e2=e4e2e3=e5−e6e2e4=e6e3e1=e4 e3e2=e5e3e3=e6e4e1=e5+e6e4e2=e6e5e1=e6 Summarizing the results of the present section and using the information from Tables B and C from Appendix, we have the following theorem. Theorem 7. Let Nbe a 6-dimensional complex one-generated nilpotent Novikov algebra. Then Nis isomorphic to N6 j,where j= 01,...,38 (see Table D presented in Appendix). APPENDIX Table A. The list of cohomology spaces of 4-dimensional one-generated nilpotent Novikov algebras. 20 Z2N4 01=∆11,∆12,∆21,∆13,∆14 −∆41,∆22 + ∆31 + 2∆41  B2N4 01=∆11,∆12,∆21  H2N4 01=[∆13],[∆14]−[∆41],[∆22] + [∆31] + 2[∆41] Z2N4 02=∆11,∆12,∆13 −∆31,∆21,2∆14 −∆23 −∆41  B2N4 02=∆11,∆13 −∆31,∆21  H2N4 02=[∆12],2[∆14]−[∆23]−[∆41] Z2N4 03(λ)λ6∈{0,1}=∆11,∆12,∆21,(2 −λ)∆13 +λ∆22 +λ∆31, (3 −2λ)∆14 + (2 −λ)λ∆23 +λ∆32 +λ∆41  B2N4 03(λ)λ6∈{0,1}=∆11,∆12 +λ∆21,(2 −λ)∆13 +λ∆22 +λ∆31  H2N4 03(λ)λ6∈{0,1}=[∆21],(3 −2λ)∆14 + (2 −λ)λ∆23 +λ∆32 +λ∆41  Z2N4 03(0)=∆11,∆12,∆13,∆14,∆21,2∆23 −2∆32 + ∆41  B2N4 03(0)=∆11,∆12,∆13  H2N4 03(0)=[∆14],[∆21],2[∆23]−2[∆32] + [∆41] Z2N4 03(1)=∆11,∆12,∆21,∆13 + ∆22 + ∆31,∆14 + ∆23 + ∆32 + ∆41  B2N4 03(1)=∆11,∆12 + ∆21,∆13 + ∆22 + ∆31  H2N4 03(1)=[∆21],[∆14] + [∆23] + [∆32] + [∆41] Z2N4 04=∆11,∆12,∆13,∆21,2∆14 + ∆22 + ∆31,∆14 −2∆23 + 2∆32 −∆41  B2N4 04=∆11,∆12,2∆13 + ∆21  H2N4 04=[∆13],2[∆14] + [∆22] + [∆31],[∆14]−2[∆23] + 2[∆32]−[∆41] Z2N4 05=∆11,∆12,∆21,∆13 + ∆22 + ∆31,−2∆13 + ∆14 + ∆23 + ∆32 + ∆41  B2N4 05=∆11,∆12 + ∆21,∆13 + ∆21 + ∆22 + ∆31  H2N4 05=[∆21],−2[∆13] + [∆14] + [∆23] + [∆32] + [∆41] Table B. The list of 5-dimensional one-generated nilpotent Novikov algebras. N5 01 :e1e1=e2e1e2=e4e1e3=e5e2e1=e3e3e1=−e5 N5 02(λ):e1e1=e2e1e2=e3e1e3= (2 −λ)e5 e2e1=λe3+e4e2e2=λe5e3e1=λe5 N5 03(λ, µ):e1e1=e2e1e2=e3e1e3=λe5e1e4=µe5 e2e1=e4e2e2=e5e3e1=e5e4e1= (2 −µ)e5 N5 04(λ)λ6=0 :e1e1=e2e1e2=e3e1e3=e5e1e4=λe5 e2e1=e4e4e1=−λe5 N5 05 :e1e1=e2e1e3=e4e1e4= 2e5e2e1=e3 e2e3=−e5e3e1=−e4e4e1=−e5 N5 06 :e1e1=e2e1e2=e5e1e3=e4e1e4= 2e5 e2e1=e3e2e3=−e5e3e1=−e4e4e1=−e5 N5 07(λ):e1e1=e2e1e2=e3e1e3= (2 −λ)e4e1e4= (3 −2λ)e5e2e1=λe3 e2e2=λe4e2e3=λ(2 −λ)e5e3e1=λe4e3e2=λe5e4e1=λe5 N5 08 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e1=e5 N5 09(λ):e1e1=e2e1e2=e3e1e3= 2e4e1e4=λe5 e2e3= 2e5e3e2=−2e5e4e1=e5 N5 10 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e2e1=e3+e5 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 N5 11(λ):e1e1=e2e1e2=e3e1e3= 2e4+λe5e1e4= 2e5 e2e1=e4e2e2=e5e3e1=e5 N5 12(λ):e1e1=e2e1e2=e3e1e3= 2e4e1e4= (2λ+ 1)e5e2e1=e4 21 e2e2=λe5e2e3=−2e5e3e1=λe5e3e2= 2e5e4e1=−e5 N5 13(λ):e1e1=e2e1e2=e3e1e3=e4−2e5e1e4=e5e2e1=e3+e4+λe5 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 N5 04(0) ∼ =N5 02(0). Table C. The list of cohomology spaces of 5-dimensional one-generated nilpotent Novikov algebras. Z2N5 01=∆11,∆12,∆13 −∆31,∆14,∆21,2∆13 + ∆22 + ∆41,2∆15 −∆23 −∆51  B2N5 01=∆11,∆12,∆13 −∆31,∆21  H2N5 01=[∆14],2[∆13] + [∆22] + [∆41],2[∆15]−[∆23]−[∆51] Z2N5 02(λ)λ6=0=∆11,∆12,∆21,∆14 −∆41,∆13 −λ∆41,∆22 + ∆31 + (2 −λ)∆41, (3 −2λ)∆15 +λ(2 −λ)∆23 +λ∆32 +λ∆51  B2N5 02(λ)λ6=0=∆11,∆12,∆21,(2 −λ)∆13 +λ∆22 +λ∆31  H2N5 02(λ)λ6=0=[∆14]−[∆41],[∆13]−λ[∆41],(3 −2λ)[∆15] + λ(2 −λ)[∆23] + λ[∆32] + λ[∆51] Z2N5 02(0)=∆11,∆12,∆21,∆14 −∆41,∆13,2∆14 + ∆22 + ∆31,∆15,2∆23 −2∆32 + ∆51  B2N5 02(0)=∆11,∆12,∆21,∆13  B2N5 02(0)=[∆14]−[∆41],2[∆14] + [∆22] + [∆31],[∆15],2[∆23]−2[∆32] + [∆51] Z2N5 03(λ, µ)=*∆11,∆12,∆21,∆13,∆14 −∆41,∆22 + ∆31 + 2∆41,(2µ−1)∆15 +λ(2 −µ)∆23+ µ(2 −µ)∆24 +3−(λ+ 1)(2 −µ)∆32 + (2 −µ)∆42 + (2 −µ)∆51 + B2N5 03(λ, µ)=∆11,∆12,∆21, λ∆13 +µ∆14 + ∆22 + ∆31 + (2 −µ)∆41  H2N5 03(λ, µ)=*[∆13],[∆14]−[∆41],(2µ−1)[∆15] + λ(2 −µ)[∆23] + µ(2 −µ)[∆24]+ 3−(λ+ 1)(2 −µ)[∆32] + (2 −µ)[∆42] + (2 −µ)[∆51]+ Z2N5 04(λ)λ6=0=∆11,∆12,∆13,∆21,∆14 −∆41,∆22 + ∆31 + 2∆41,−2∆15 + ∆23 +λ∆24 −∆32 + ∆51  B2N5 04(λ)λ6=0=∆11,∆12,∆21,∆13 +λ(∆14 −∆41) H2N5 04(λ)λ6=0=[∆14]−[∆41],[∆22] + [∆31] + 2[∆41]−2[∆15] + [∆23] + λ[∆24]−[∆32] + [∆51] Z2N5 04(0)=∆11,∆12,∆13,∆15,∆21,∆14 −∆41,2∆14 + ∆22 + ∆31,2∆23 −2∆32 + ∆51  B2N5 04(0)=∆11,∆12,∆13,∆21  H2N5 04(0)=[∆15],[∆14]−[∆41],2[∆14] + [∆22] + [∆31],2[∆23]−2[∆32] + [∆51] Z2N5 05=∆11,∆12,∆13 −∆31,2∆14 −∆23 −∆41,∆21,3∆15 −2∆24 + ∆33 −∆51  B2N5 05=∆11,∆21,∆13 −∆31,2∆14 −∆23 −∆41  H2N5 05=[∆12],3[∆15]−2[∆24] + [∆33]−[∆51] Z2N5 06=∆11,∆12,∆13 −∆31,∆21,2∆14 −∆23 −∆41,3∆15 −∆22 −2∆24 −2∆31 + ∆33 −∆51  B2N5 06=∆11,∆21,∆13 −∆31,∆12 + 2∆14 −∆23 −∆41  H2N5 06=[∆12],3[∆15]−[∆22]−2[∆24]−2[∆31] + [∆33]−[∆51] Z2N5 07(λ)λ6=0=∆11,∆12,∆21,(2 −λ)∆13 +λ∆22 +λ∆31,(3 −2λ)∆14 +λ(2 −λ)∆23 +λ∆32 +λ∆41, (4 −3λ)∆15 +λ(3 −2λ)∆24 +λ(2 −λ)∆33 +λ∆42 +λ∆51  B2N5 07(λ)λ6=0=∆11,∆12 +λ∆21,(2 −λ)∆13 +λ∆22 +λ∆31,(3 −2λ)∆14 +λ(2 −λ)∆23 +λ∆32 +λ∆41  H2N5 07(λ)λ6=0=[∆21],(4 −3λ)[∆15] + λ(3 −2λ)[∆24] + λ(2 −λ)[∆33] + λ[∆42] + λ[∆51] Z2N5 07(0)=∆11,∆12,∆13,∆14,∆15,∆21,2∆23 −2∆32 + ∆41  B2N5 07(0)=∆11,∆12,∆13,∆14  H2N5 07(0)=[∆15],[∆21],2[∆23]−2[∆32] + [∆41] Z2N5 08=∆11,∆12,∆13,∆14,∆21,2∆15 + ∆22 + ∆31,∆23 −∆32 + ∆41  B2N5 08=∆11,∆12,∆13,∆14 + ∆21  H2N5 08=[∆21],2[∆15] + [∆22] + [∆31],[∆23]−[∆32] + [∆41] 22 Z2N5 09(λ)=∆11,∆12,∆13,∆14,∆21,2∆23 −2∆32 + ∆41  B2N5 09(λ)=∆11,∆12,∆13, λ∆14 + 2∆23 −2∆32 + ∆41  H2N5 09(λ)=[∆14],[∆21] Z2N5 10=∆11,∆12,∆21,∆13 + ∆22 + ∆31,∆14 + ∆23 + ∆32 + ∆41, −2∆13 + ∆15 + ∆24 + ∆33 + ∆42 + ∆51  B2N5 10=∆11,∆12 + ∆21,∆13 + ∆22 + ∆31,∆14 + ∆21 + ∆23 + ∆32 + ∆41  H2N5 10=[∆12],−2[∆13] + [∆15] + [∆24] + [∆33] + [∆42] + [∆51] Z2N5 11(λ)=∆11,∆12,∆13,∆15 + ∆32,∆21,2∆14 + ∆22 + ∆31,∆14 −2∆23 + 2∆32 −∆41  B2N5 11(λ)=D∆11,∆12,2∆13 + ∆21, λ∆13 + 2∆14 + ∆22 + ∆31 E H2N5 11(λ)=[∆13],[∆15] + [∆32],[∆14]−2[∆23] + 2[∆32]−[∆41] Z2N5 12(λ)=∆11,∆12,∆13,∆21,∆14 −2∆23 + 2∆32 −∆41,∆22 + ∆31 + 2∆14  B2N5 12(λ)=∆11,∆12,2∆13 + ∆21,(1 + 2λ)∆14 +λ∆22 −2∆23 +λ∆31 + 2∆32 −∆41  H2N5 12(λ)=[∆13],[∆14]−2[∆23] + 2[∆32]−[∆41] Z2N5 13(λ)=D∆11,∆12,∆21,∆13 + ∆22 + ∆31,−2∆13 + ∆14 + ∆23 + ∆32 + ∆41, (1 −2λ)∆13 −2∆14 + ∆15 −∆23 + ∆24 + ∆33 + ∆41 + ∆42 + ∆51 E B2N5 13(λ)=∆11,∆12 + ∆21,∆13 + ∆21 + ∆22 + ∆31,−2∆13 + ∆14 +λ∆21 + ∆23 + ∆32 + ∆41  H2N5 13(λ)=[∆12],(1 −2λ)[∆13]−2[∆14] + [∆15]−[∆23] + [∆24] + [∆33] + [∆41] + [∆42] + [∆51] Table D. The list of 6-dimensional one-generated nilpotent Novikov algebras N6 01 :e1e1=e2e1e2=e3e1e3=e5e1e4=e6 e2e1=e4e4e1=−e6 N6 02(λ):e1e1=e2e1e2=e3e1e3=e5e1e4=λe6 e2e1=e4e2e2=e6e3e1=e6e4e1= (2 −λ)e6 N6 03(λ, µ):e1e1=e2e1e2=e3e1e3=λe5+µe6e1e4=e5 e2e1=e4e2e2=e6e3e1=e6e4e1=−e5+ 2e6 N6 04 :e1e1=e2e1e2=e5e1e3=e4e1e4= 2e6 e2e1=e3e2e3=−e6e3e1=−e4e4e1=−e6 N6 05(λ):e1e1=e2e1e2=e3e1e3= (2 −λ)e4e1e4= (3 −2λ)e5e2e1=λe3+e6 e2e2=λe4e2e3=λ(2 −λ)e5e3e1=λe4e3e2=λe5e4e1=λe5 N6 06(λ):e1e1=e2e1e2=e3e1e3= 2e4e1e4=λe5e2e1=e6 e2e3= 2e5e3e2=−2e5e4e1=e5 N6 07 :e1e1=e2e1e2=e3e1e3= 2e4e1e4=e5 e2e1=e5e2e3= 2e6e3e2=−2e6e4e1=e6 N6 08 :e1e1=e2e1e2=e3e1e3= 2e4e1e4=e5 e2e3= 2e6e3e2=−2e6e4e1=e6 N6 09 :e1e1=e2e1e2=e3e1e3= 2e4+e6e1e4= 2e5 e2e1=e4e2e2=e5e3e1=e5 N6 10(λ):e1e1=e2e1e2=e3e1e3= 2e4+e6e1e4= (2λ+ 1)e5e2e1=e4 e2e2=λe5e2e3=−2e5e3e1=λe5e3e2= 2e5e4e1=−e5 N6 11(λ):e1e1=e2e1e2=e3e1e3= 2e4+λe6e1e4=e5+ 2e6e2e1=e4 e2e2=e6e2e3=−2e5e3e1=e6e3e2= 2e5e4e1=−e5 N6 12 :e1e1=e2e1e2=e3e1e3=e4−2e5e1e4=e5e2e1=e3+e4+e6 e2e2=e4e2e3=e5e3e1=e4e3e2=e5e4e1=e5 N6 13 :e1e1=e2e1e2=e4e1e3=e5e1e4=e6e1e5= 2e6 e2e1=e3e2e3=−e6e3e1=−e5e5e1=−e6 N6 14(α):e1e1=e2e1e2=e3e1e3=e5+e6e1e4=αe6 e1e5=e6e2e1=e3+e4e2e2=e5e2e3=e6 e3e1=e5e3e2=e6e4e1=−(α+ 1)e6e5e1=e6 N6 15 :e1e1=e2e1e2=e3e1e3=−e5+e6e1e5=−3e6 e2e1= 3e3+e4e2e2= 3e5e2e3=−3e6e3e1= 3e5 e3e2= 3e6e4e1=−3e6e5e1= 3e6 N6 16(λ)λ6=3 :e1e1=e2e1e2=e3e1e3= (2 −λ)e5e1e4=e6 e1e5= (3 −2λ)e6e2e1=λe3+e4e2e2=λe5e2e3=λ(2 −λ)e6 e3e1=λe5e3e2=λe6e4e1=−e6e5e1=λe6 N6 17(α):e1e1=e2e1e2=e3e1e3= 2e5e1e4=e6 e1e5=αe6e2e1=e4e2e3= 2e6e3e1=e6 e3e2=−2e6e4e1=−e6e5e1=e6 N6 18(α):e1e1=e2e1e2=e3e1e3= 2e5e1e4= (α+ 2)e6e1e5=e6 e2e1=e4e2e2=e6e3e1=e6e4e1=−αe6 N6 19(α, β):e1e1=e2e1e2=e3e1e3= 2e5e1e4= (α+ 2)e6 e1e5=βe6e2e1=e4e2e2=e6e2e3= 2e6 e3e1=e6e3e2=−2e6e4e1=−αe6e5e1=e6 23 N6 20(λ, µ):e1e1=e2e1e2=e3e1e3=λe5e1e4=µe5e1e5= (2µ−1)e6 e2e1=e4e2e2=e5e2e3=λ(2 −µ)e6e2e4=µ(2 −µ)e6 e3e1=e5e4e1= (2 −µ)e5e3e2=3−(λ+ 1)(2 −µ)e6e4e2= (2 −µ)e6e5e1= (2 −µ)e6 N6 21(λ, µ)µ6=2 :e1e1=e2e1e2=e3e1e3=λe5+e6e1e4=µe5e1e5= (2µ−1)e6 e2e1=e4e2e2=e5e2e3=λ(2 −µ)e6e2e4=µ(2 −µ)e6 e3e1=e5e4e1= (2 −µ)e5e3e2=3−(λ+ 1)(2 −µ)e6e4e2= (2 −µ)e6e5e1= (2 −µ)e6 N6 22(λ):e1e1=e2e1e2=e3e1e3=λe5e1e4=e5+e6e1e5=e6 e2e1=e4e2e2=e5e2e3=λe6e2e4=e6 e3e1=e5e4e1=e5−e6e3e2= (2 −λ)e6e4e2=e6e5e1=e6 N6 23(ν)ν6=0 :e1e1=e2e1e2=e3e1e3=e5+νe6e1e4=e5+e6e1e5=e6 e2e1=e4e2e2=e5e2e3=e6e2e4=e6 e3e1=e5e4e1=e5−e6e3e2=e6e4e2=e6e5e1=e6 N6 24(λ):e1e1=e2e1e2=e3e1e3=λe5e1e4= 2e5+e6e1e5=e6 e2e1=e4e2e2=e5e3e1=e5e3e2=e6 N6 25(λ)λ6=0 :e1e1=e2e1e2=e3e1e3=e5e1e4=λe5e1e5=−2e6 e2e1=e4e2e3=e6e2e4=λe6e3e2=−e6 e4e1=−λe5e5e1=e6 N6 26(λ)λ6=0 :e1e1=e2e1e2=e3e1e3=e5e1e4=λe5+e6e1e5=−2e6 e2e1=e4e2e3=e6e2e4=λe6e3e2=−e6 e4e1=−λe5−e6e5e1=e6 N6 27 :e1e1=e2e1e3=e4e1e4= 2e5e1e5= 3e6 e2e1=e3e2e3=−e5e2e4=−2e6e3e1=−e4 e3e3=e6e4e1=−e5e5e1=−e6 N6 28 :e1e1=e2e1e2=e5e1e3=e4e1e4= 2e5e1e5= 3e6 e2e1=e3e2e2=−e6e2e3=−e5e2e4=−2e6e3e1=−e4−2e6 e3e3=e6e4e1=−e5e5e1=−e6 N6 29(λ):e1e1=e2e1e2=e3e1e3= (2 −λ)e4e1e4= (3 −2λ)e5e1e5= (4 −3λ)e6 e2e1=λe3e2e2=λe4e2e3=λ(2 −λ)e5e2e4=λ(3 −2λ)e6e3e1=λe4 e3e2=λe5e3e3=λ(2 −λ)e6e4e1=λe5e4e2=λe6e5e1=λe6 N6 30 :e1e1=e2e1e2=e3e1e4=−e5e1e5=−2e6 e2e1= 2e3+e6e2e2= 2e4e2e4=−e6e3e1= 2e4 e3e2= 2e5e4e1= 2e5e4e2= 2e6e5e1= 2e6 N6 31 :e1e1=e2e1e2=e3e1e3= 2e4e1e4= 3e5 e1e5=e6e2e3= 2e6e3e2=−2e6e4e1=e6 N6 32 :e1e1=e2e1e2=e3e1e3= 2e4 e1e4= 3e5e1e5=e6e2e1=e6 N6 33 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5e1e5=e6 e2e1=e3+e6e2e2=e4e2e3=e5e2e4=e6e3e1=e4 e3e2=e5e3e3=e6e4e1=e5e4e2=e6e5e1=e6 N6 34(λ):e1e1=e2e1e2=e3e1e3=e4e1e4=e5 e1e5= 2e6e2e1=e5+λe6e2e2=e6e3e1=e6 N6 35(λ)λ6=0 :e1e1=e2e1e2=e3e1e3=e4e1e4=e5 e1e5= 2e6e2e1=e5e2e2=e6e2e3=λe6 e3e1=e6e3e2=−λe6e4e1=λe6 N6 36 :e1e1=e2e1e2=e3e1e3=e4−2e6e1e4=e5e1e5=e6 e2e1=e3+e5e2e2=e4e2e3=e5e2e4=e6e3e1=e4 e3e2=e5e3e3=e6e4e1=e5e4e2=e6e5e1=e6 N6 37(λ, µ):e1e1=e2e1e2=e3e1e3= 2e4+λe5e1e4= 2e5+µe6 e1e5=e6e2e1=e4e2e2=e5e2e3=−2µe6 e3e1=e5e3e2= (1 + 2µ)e6e4e1=−µe6 N6 38(λ):e1e1=e2e1e2=e3e1e3=e4−2e5+ (1 −2λ)e6e1e4=e5−2e6e1e5=e6 e2e1=e3+e4+λe5e2e2=e4e2e3=e5−e6e2e4=e6e3e1=e4 e3e2=e5e3e3=e6e4e1=e5+e6e4e2=e6e5e1=e6 N6 16(3) ∼ =N6 05(3),N6 21(λ, 2) ∼ =N6 20(λ, 2),N6 23(0) ∼ =N6 22(1),N6 25(0) ∼ =N6 06(−2), N6 26(0) ∼ =N6 17(−2),N6 35(0) ∼ =N6 34(0),N6 34(λ)∼ =N6 34(−λ),N6 35(λ)∼ =N6 35(−λ). 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